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Dynamic generalized linear models for monitoring endemic diseasesmoving beyond univariate process monitoring control algorithms

Lopes Antunes, Ana Carolina; Jensen, Dan; Hisham Beshara Halasa, Tariq; Toft, Nils

Published in:SVEPM 2016 conference proceedings

Publication date:2016

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Lopes Antunes, A. C., Jensen, D., Hisham Beshara Halasa, T., & Toft, N. (2016). Dynamic generalized linearmodels for monitoring endemic diseases: moving beyond univariate process monitoring control algorithms. InSVEPM 2016 conference proceedings

DYNAMIC GENERALIZED LINEAR MODELS FOR MONITORING ENDEMIC

DISEASES: MOVING BEYOND UNIVARIATE PROCESS MONITORING CONTROL

ALGORITHMS

A.C. LOPES ANTUNES*, D. JENSEN, T. HALASA, N. TOFT

SUMMARY

The objective was to use a Dynamic Generalized Linear Model (DGLM) based on a

binomial distribution with a linear trend, for monitoring the PRRS (Porcine Reproductive and

Respiratory Syndrome sero-prevalence in Danish swine herds. The DGLM was described and

its performance for monitoring control and eradication programmes based on changes in

PRRS sero-prevalence was explored. Results showed a declining trend in PRRS sero-

prevalence between 2007 and 2014 suggesting that Danish herds are slowly eradicating

PRRS. The simulation study demonstrated the flexibility of DGLMs in adapting to changes in

trends in sero-prevalence. Based on this, it was possible to detect variations in the growth

model component. This study is a proof-of-concept, demonstrating the use of DGLMs for

monitoring endemic diseases. In addition, the principles stated might be useful in general

research on monitoring and surveillance of endemic and (re-)emerging diseases.

INTRODUCTION

New methods for monitoring animal diseases continue to be an active area of research. In

the past decade, several studies applied statistical quality control methods for syndromic

surveillance in human and veterinary medicine (Buckeridge et al., 2005; Jackson et al., 2007;

Dórea et al., 2013). Many of these studies applied univariate process monitoring control

algorithms to detect outbreaks of re-emerging diseases. In these cases, control and/or

eradication measures are implemented whenever certain threshold levels related to the

infection or disease status have been exceeded. However, the term “monitoring” can also be

used to describe actions, where a continuous process of collecting data on animal diseases is

ongoing, but without any instant control activities (Salman, 2003).

For endemic diseases, it is common to implement control and eradication programmes at

herd and regional levels to reduce the economic impact of diseases. Often, these programmes

are based on laboratory diagnostics. One example is the Danish monitoring programme for

Porcine Reproductive and Respiratory Syndrome (PRRS).

* Ana Carolina Lopes Antunes, National Veterinary Institute, Technical University of

Denmark, Bülowsvej 27, 1870 Frederiksberg C, Denmark. Email: aclan@vet.dtu.dk

Despite disease control efforts in Denmark, PRRS continues to contribute towards the

economic losses of the industry since its first diagnosis in 1992. PRRS monitoring is

primarily based on serological testing performed on regular basis from herds that have the

Specific Pathogen Free System (SPF) certificate (Specific Pathogen Free System (SPF-SuS),

2015). The frequency of testing depends on the SPF herd type, being performed once a month

for breeding herds and once a year for finisher herds. The SPF herds represent about 40% of

all Danish swine (SPF-SuS, 2015). For non-SPF herds, PRRS diagnostic test are not

mandatory and different reasons might explain the variation in frequency of laboratory

testing. Thus, diagnostic laboratory submissions of PRRS are collected based on different

purposes and frequencies in Denmark.

For disease monitoring, the resulting time series are characterized by observational noise

as a result of the variation in the disease prevalence and of the number of samples and herds

tested over time. Furthermore, its randomness and non-stationary nature are difficult to

model. In these cases, it is necessary to use models with a more dynamic structure, where it is

possible to add trends, cyclic patterns and also allow the parameters to change over time.

State space models are one possible approach in which relevant prior knowledge and current

information are combined. While state space models have been adopted in herd management

(Jensen et al., 2015; Madsen & Kristensen, 2005; Ostersen et al., 2010), their use has been

underutilized in veterinary sciences for diseases surveillance purposes. In the literature, there

are few studies using these type of models for disease monitoring and surveillance in humans

(Cao et al., 2014; Cowling et al., 2006).

The objective was to use a state space model for monitoring the PRRS sero-prevalence in

Danish swine herds. The binomial DGLM with a linear growth was described and its

performance for monitoring control and eradication programmes based on changes in PRRS

sero-prevalence was explored. This study is a proof of concept, demonstrating the use of

DGLMs for monitoring endemic disease, but the principles stated might also be useful in

general research on monitoring and surveillance of endemic and (re-)emerging diseases.

MATERIALS AND METHODS

Data source

Laboratory submission data stored in the National Veterinary Institute – Technical

University of Denmark (DTU Vet) information management system and in the Laboratory for

Swine Diseases-SEGES Pig Research Centre (VSP-SEGES) were used to determine the

weekly PRRS sero-prevalence in Danish swine herds from January 2007 to December 2014.

Each laboratory submission consisted of individual blood samples collected from the same

herd on the same day from different animals. Only submissions where at least 2 individual

blood samples were tested by serological tests including Blocking Enzyme-Linked

Immunosorbent Assay (ELISA) and/or Immunoperoxidase monolayer assay (IPMA) for one

or both PRRSV (Porcine Reproductive and Respiratory Syndrome Virus)strains were

included in the analysis. These serological tests used were a DTU Vet “in-house” ELISA

(Sørensen et. al, 1997) and IPMA (Bøtner et al., 1994). Furthermore, diagnostic test results

performed at VSP-SEGES were based on IDEXX PRRS X3 Ab ELISA test (IDEXX,

Ludwigsburg, Germany). Results from experimental studies were excluded from the analysis.

Herds were classified as PRRS sero-positive when at least 2 individual blood samples in

each submission tested PRRS positive, independently of the PRRS strain. The between-herd

PRRS sero-prevalence was calculated weekly as the proportion of PRRS positive herds from

the total number of herds tested for PRRS.

Modelling

A binomial DGLM with a linear growth as described by West and Harrison (1997) was

used to model the data. The general purpose of the DGLM is to estimate the underlying

parameter vector from the observed data (𝛳) combined with any prior information available

at time 0 (D0) before any observation is made. This can be achieved sequentially where the

estimated value is updated each time a new value (PRRS sero-prevalence) is obtained. In this

case, the conditional distribution of 𝛳𝑡 given by D𝑡 (𝛳𝑡|D𝑡) was estimated. These models can

be used to estimate a one-step forecast of the mean, allowing for a comparison with the actual

observed PRRS sero-prevalence. Moreover, the linear growth component includes a time-

varying slope (or local linear trend), allowing the system to adapt to a possible positive or

negative growth for each t.

In general, the DGLM consists of an observation equation (Eq. 1) and a system equation

(Eq. 2):

ɡ(𝑝𝑡) = 𝐹𝑡′𝛳𝑡 (1)

𝛳𝑡 = 𝐺𝑡𝛳𝑡−1 + 𝑊𝑡 (2)

Equation 1 describes how the values of an observation (PRRS sero-prevalence) derive

from ɡ(𝑝𝑡), depends on an unobservable parameter vector (𝛳) for time t based on a linear

function. For the model specification, ɡ() is the identity function. Equation 2 describes the

dynamic properties of the unobservable parameter vector 𝛳. In this study, the transposed

design matrix (𝐹𝑡′) has the structure presented in Table 1, in order to estimate underlying

values of PRRSV sero-prevalence according to Eq 1. The system matrix (𝐺) used to update

the mean of the PRRSV sero-prevalence for each time step taking into account the trend.

Both matrix structures were constant for each t (week). The variance-covariance matrix (𝑊𝑡)

describes the evolution of variance and covariance of each parameter for each time step.

Rather than estimating (𝑊𝑡), the system variance was modelled using a discount factor (see

Eq. 4).

Table 1. Matrices structure used in Eq. 1 and 2.

𝐹𝑡′ 𝐺𝑡

[1 0] [1 1

0 1]

The DGLM update for each time step 𝑡 was performed as follows:

a) the posterior distribution for 𝛳𝑡−1 was expressed by a prior mean (𝑚𝑡−1) and a

variance (𝐶𝑡−1), (𝛳𝑡−1 |𝐷𝑡−1)~[𝑚𝑡−1, 𝐶𝑡−1];

b) the prior distribution for 𝛳𝑡 (𝛳𝑡 |𝐷𝑡−1)~[𝑎𝑡, 𝑅𝑡] was made based on the prior mean

(𝑎𝑡) and prior variance (𝑅𝑡) which were calculated as described in Eq. 3 and 4. The

specification of the variance components was specified using a discount factor (𝛿);

𝑎𝑡 = 𝐺𝑡𝑚𝑡−1 (3)

𝑅𝑡 =

1

𝛿𝐺𝑡𝐶𝑡−1𝐺𝑡

′ (4)

c) the prior distribution for 𝑌𝑡 (𝑌𝑡 |𝐷𝑡−1)~[𝑓𝑡, 𝑞𝑡] was calculated based on the forecast

mean (𝑓𝑡) and forecast variance (𝑞𝑡) (Eq. 5 and 6);

𝑓𝑡 = 𝐹𝑡′𝑎𝑡 (5)

𝑞𝑡 = 𝐹𝑡′𝑅𝑡𝐹𝑡 (6)

d) the posterior mean (𝑓𝑡∗) and variance (𝑞𝑡

∗) were calculated as described in Eq. 7 and 8.

In this case, it was assumed that the prior probability 𝑝 (PRRS sero-prevalence) of a

binomial distribution was Beta(𝛼, 𝛽). If 𝜅 successes (PRRS positive herds) out of 𝑛

trials (number of herds tested for PRRS) were observed, the posterior p, given the new

observation was Beta(𝛼𝑡 + 𝜅𝑡 , 𝛽𝑡 + 𝑛𝑡 − 𝜅𝑡). The parameters 𝛼𝑡 and 𝛽𝑡 were

calculated according to Eq. 9 and 10.

𝑓𝑡

∗ = 𝛼𝑡 + 𝜅𝑡

𝛼𝑡 + 𝛽𝑡 + 𝑛𝑡

(7)

𝑞𝑡

∗ = 𝑓𝑡

∗(1 − 𝑓𝑡∗)

𝛼𝑡 + 𝛽𝑡 + 𝑛𝑡 + 1

(8)

𝛼𝑡 = 𝑓𝑡 (

𝑓𝑡(1 − 𝑓𝑡)

𝑞𝑡− 1)

(9)

𝛽𝑡 = (1 − 𝑓𝑡) (

𝑓𝑡(1 − 𝑓𝑡)

𝑞𝑡− 1)

(10)

e) the posterior distribution for 𝛳𝑡−1 in a) was calculated based on its mean matrix 𝑚𝑡

and its variance-covariance matrix 𝐶𝑡 as demonstrated in Eq. 11 and 12.

𝑚𝑡 = 𝑎𝑡 + 𝑅𝑡𝐹𝑡(𝑓𝑡∗ − 𝑓𝑡)/𝑞𝑡 (11)

𝐶𝑡 = 𝑅𝑡 − 𝑅𝑡𝐹𝑡𝐹𝑡′𝑅𝑡(1 − 𝑞𝑡

∗ 𝑞𝑡⁄ )/𝑞𝑡 (12)

Model initialization: Reference analysis was used to estimate the initial parameters

𝐷0~[𝑚0, 𝐶0] as described by West and Harrison (1997). We defined the matrices 𝐾𝑡 and 𝐻𝑡

and the vectors 𝑘𝑡 and ℎ𝑡 for the first two observations 𝑝1:2.

For 𝑡 = 1, the initial parameters were defined as 𝐻1=0, ℎ1=0, 𝐾1 = 𝐻1+ 𝐹1𝐹1′ and 𝑘1 = ℎ1 +

𝐹1𝑝1. For 𝑡 = 2, the vectors and matrices were updated as described in Eq. 13 to Eq. 16.

𝐻2 = 𝐺2−1′𝐾1𝐺2

−1 (13)

ℎ2 = 𝐺2−1′𝑘1 (14)

𝐾2 = 𝐻2+ 𝐹2𝐹2′ (15)

𝑘2 = ℎ2 + 𝐹2𝑝2 (16)

Then, the prior distribution for 𝑡 = 3 was calculated according to Eq. 17 and 18.

𝑚2 = 𝐾2−1𝑘2 (17)

𝐶2 = 𝐾2−1 (18)

System variance: The DGLM model was run based on different discount factors (𝛿) ranging

from 0.1 up to 1 by increments of 0.01. The discount factor which minimized the sum of the

squared forecast errors based on the first two years of the data was chosen for the analysis.

Monitoring model components: The values obtained from the 𝑚 vector for each time step t

were used to decompose the time series and obtain the model growth (PRRS sero-prevalence

trend). The variance on the growth parameter was calculated from the 𝐶 matrix and used to

calculate 95% confidence intervals (CI).

Simulated scenarios: PRRS sero-prevalence baseline was simulated for 8 years, in which the

number of positive herds (X) per week was drawn from a binomial distribution

(𝑋 ~ 𝑏𝑖𝑛(n, p)) with a probability (p) (PRRS sero-prevalence) and a sample size (n) equal

to the number of Danish herds tested for PRRS per week between 2007 and 2014. The

weekly sero-prevalence was calculated as the simulated number of sero-positive herds

divided by the total weekly number of herds tested. The first 104 weeks were simulated with

a constant initial prevalence of 0.24, corresponding to the average PRRS sero-prevalence in

Danish herds observed based on the laboratory diagnostic data from 2007 to 2014. In the first

scenario (Scenario A), a constant decrease from p=0.24 to p=0.10 during 4 years followed by

constant sero-prevalence was simulated. The second scenario (Scenario B) represented a

decrease in the sero-prevalence from p=0.24 to p=0.10 during 2 years, followed by an

increase to p=0.18 during the subsequent 2 years.

The sensitivity (Se) and timeliness were used to evaluate the performance of the DGLM to

detect significant changes in the simulated scenarios. The Se was defined as the proportion of

simulations in which significant changes in the model growth component from zero were

found. Timeliness was defined as the number of weeks between a change in the PRRS sero-

prevalence (decrease, increase, constant) was simulated and detected.

Convergence rate: A total of 20,000 simulations of weekly PRRS sero-prevalence with a

constant decrease from 0.24 to 0.05 over 5 years were carried out. The number of iterations

needed to reach a stable variance in the average time to detect significant changes

(convergence) was determined visually by plotting the variance of the average timeliness

with a stepwise increase of 100 iterations up to 20,000 iterations against the number of

iterations. Stable results were observed when using only 10,000 iterations and hence all

further simulations were run with 10,000 iterations.

All analyses were performed using R (version 3.1.1) (R Core Team, 2014).

RESULTS

Data description

A total of 56,341 laboratory submissions from 5,390 Danish swine were included in the

analysis. The average weekly number of herds tested for PRRSV was 130 (min=9,

max=206); the mean weekly number of PRRS positive herds was 31 herds (min=0, max=

60). The weekly average PRRS sero-prevalence was 0.24 (min=0, max=0.38). The yearly

average of PRRS sero-prevalence declined from 0.28 in 2007 to 0.20 in 2014, with an

average decrease of 0.01 per year.

Model initialization and discount factor

Table 2 shows the posterior 𝐶2 and 𝑚2 matrices obtained from the reference analysis and

used as priors for the DGLM model for 𝑡 = 3. The discount factor which minimized the sum

of forecast errors for the data was 𝛿=0.98.

Table 2. Priors for t = 3 obtained from the reference analysis.

𝑚2 𝐶2

[0.30

0.07] [

1 1

1 2]

Modelling and decomposing DGLM

Results show a declining trend of PRRS sero-prevalence between 2007 and 2014.

Significant decreases (95% CI excluding zero) were detected mainly in the last 6 months of

2007; end of 2008 to the first semester of 2010 and from the last quarter of 2010 until the

beginning of 2013 (Fig. 1). No significant increases in PRRS sero-prevalence were observed

and all values for the growth component were below 0.

Fig. 1 Using a DGLM to monitor PRRS sero-prevalence in Danish swine herds from 2007 to

2014. Results show the weekly PRRS sero-prevalence and the filtered mean (black) (a) and

the corresponding DGLM growth component (b). The black rugs indicate were the growth

component is significantly different from zero.

Simulated scenarios

The simulated scenarios are represented in Fig. 2 and 3. The results for the simulation

study are presented in Table 3. Significant changes in the model growth component from zero

were found in both scenarios. However, the DGLM detected changes in the growth with a

higher sensitivity for decreasing changes when compared to constant growth in the time

series. The lowest sensitivity was found for Scenario A when the PRRS sero-prevalence

became constant after the decrease, with the DGLM growth component being non-

significantly different from zero in 39.02% of the simulations.

Table 3. Timeliness (weeks) and Se for the simulated scenarios.

Scenario A Scenario B

Intervention Decrease Constant Decrease Increase

Timeliness (median)

(min-max)

Se (%)

47

(0-89)

100

96

(57-106)

39.02

27

(0-56)

100

146

(110-257)

99.64

a

b

Fig. 2 Simulated control program Scenario A. PRRS sero-prevalence was constant during 104

weeks, followed by a decrease to 0.10 during 208 weeks and then a constant prevalence. The

DGLM filtered mean (black line) (a) and the corresponding DGLM growth component (b)

(grey lines) are presented. The black rugs indicate a significant negative the growth

component based on 95% CI.

Fig. 3 Simulated control program Scenario B. PRRS sero-prevalence was constant during 104

weeks, followed by a decrease to 0.10 during 104 weeks and an increase up to 0.18 during

104 weeks. The DGLM filtered mean (black line) (a) and the corresponding DGLM growth

component (b) (grey lines) are presented. The black and grey rugs indicate significant

declines and increase in the growth component based on 95% CI, respectively.

a

b

a

b

DISCUSSION

The objective of this study was to use a binomial DGLM with a linear growth component

for monitoring PRRS sero-prevalence in similar contexts to the Danish Pig Industry. The

same model can be used for monitoring other prevalence data. These types of models can also

be derived for Poisson distribution for monitoring count data, such as the number of samples

submitted for analysis etc. Moreover, an ordinary Dynamic Linear Model (DLM) can be used

if the data are normally distributed. They also allow for modelling interventions as well as

changes in level shift through multi-process models (Thysen, 1993). The DGLM provide a

flexible framework in which it is possible to include different data sources in a multivariate

process as shown by (Jensen et al., 2015). Moreover, the use of this method allows

monitoring of trends and also other components of time series such as seasonal, regression

and autoregressive effects components which have a wide interest in biomedical time series

applications (West & Harrison, 1997).

As no information on PRRS outbreaks and eradication programmes is available for

Danish swine herds, a simulation study was conducted. One limitation of this study is related

to the simulation approach used; the simulated sero-prevalence was based on a binomial

distribution. The variation in the number of herds tested had an impact on the simulated

prevalence contributing to the variation (noise in the baseline). As a consequence, the

timeliness to detect interventions showed a wide range of values and the sensitivity was not

similar for all interventions. One approach to overcome this issue could be to aggregate the

data on a monthly basis, thus reducing the noise in the baseline and possibly improve the

performance of the model to adapt to changes in the trend.

The DGLM model was able to detect changes in both scenarios. However, it is important

to notice that decreases were larger compared to the increases, corresponding to an absolute

decay in sero-prevalence of 0.145 and absolute increase of 0.08. For scenario B, significant

positive changes in the model growth component were found after a period in which non-

significant changes were found. These justify the longer time needed to detect increases. The

variation in the growth parameter was monitored based on 95% CI’s. Different approaches

could be, e.g. Shewart control charts, cumulative sensitivities, V-mask (Montgomery, 2013)

or target values, which might yield improved the performances.

In a Bayesian framework the choice of priors is critical for making inference. Reference

analysis was used to initiate the DGLM model. From a practical point of view, when a system

is set up, the number of observations is low to make the influence of the priors significant. In

this case, the use of “non-informative” priors can be used. This method offers an easily

applied default analysis (West & Harrison, 1997) when running a DGLM. However, it can be

seen from the simulated scenarios that the DGLM takes 3 months to adapt to the data. For

this reason, it is important to have historical data (retrospective analysis) to train the model

when setting up a monitoring system.

The systems variance was defined based on a discount factor, expressing the decay of

information in the system. Defining 𝛿=0.98 implies a small systems variance with a very

slow adaptation to new observations. This value was defined using the same method

described in Kristensen et al. (2010), where 𝛿 should optimized for the performance of the

model in making forecasts, i.e., minimizing the forecast errors for the first two years of data

(retrospective analysis). In recent literature (Bono et al., 2012; Jensen et al., 2015), the

Expectation-Maximization algorithm (Dempster et al., 1977) was used to define the 𝑊

variance-covariance matrix. This approach offers a general approach to iterative computation

of maximum-likelihood estimates when the observations can be viewed as incomplete data.

The use of a discount factor provides a parsimonious approach when compared to the full

estimation of 𝑊.

In summary, results show a declining trend on PRRS sero-prevalence between 2007 and

2014 suggesting more Danish herds are eradicating PRRS. The simulation study highlighted

that DGLM are flexible models able to adapt to changes in the time series. It was possible to

detect variations in the growth component of simulated scenarios. This study is a proof of

concept, demonstrating the use of DGLMs for monitoring endemic disease, but the principles

stated might also be useful in general modelling, monitoring and surveillance of (re)emerging

diseases. Further analysis to compare the performance of the DGLM, including different

components, to other models will be investigated in future studies.

ACKNOWLEDGEMENTS

The authors would like to thank to VSP-SEGES for providing the serological test results.

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